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We shall define the $(\\rho, u, H)$-equivariant Picard group of $A$, which is denoted by $\\Pic_H^{\\rho, u}(A)$, and discuss basic properties of $\\Pic_H^{\\rho, u}(A)$. Also, we suppose that $(\\rho, u)$ is the coaction of $H^0$ on the unital $C^*$-algebra $A$, that is, $u=1\\otimes 1^0$. We investigate the relation between $\\Pic(A^s )$, the ordinary Picard group of $A^s$ and $\\Pic_H^{\\rho^s}(A^s )$ where $A^s$ is the stable $C^*$-"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1512.07724","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2015-12-24T06:13:05Z","cross_cats_sorted":[],"title_canon_sha256":"3445549a1961eccd4e6bb0098e93e58ace65b645b34d25e89b848814868daeea","abstract_canon_sha256":"b3269e2d7c528777d2b32c5947333db04605700a8233d15033a5b0030c700abc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:48.044508Z","signature_b64":"LjYajaw7a/hNXPVIIE8q11QLyIAO5iaG1MnPBl9Ny2Q076PyZKVNw+11uyCo+aaFOnck5UNjgRfll9zRfk85AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b40bf4f5579c3b10996b2224faebd3de63c3456db3abe12a6d4124331be90dfa","last_reissued_at":"2026-05-18T01:23:48.043826Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:48.043826Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Equivariant Picard groups of $C^*$-algebras with finite dimensional $C^*$-Hopf algebra coactions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Kazunori Kodaka","submitted_at":"2015-12-24T06:13:05Z","abstract_excerpt":"Let $A$ be a $C^*$-algebra and $H$ a finite dimensional $C^*$-Hopf algebra with its dual $C^*$-Hopf algebra $H^0$. Let $(\\rho, u)$ be a twisted coaction of $H^0$ on $A$. We shall define the $(\\rho, u, H)$-equivariant Picard group of $A$, which is denoted by $\\Pic_H^{\\rho, u}(A)$, and discuss basic properties of $\\Pic_H^{\\rho, u}(A)$. Also, we suppose that $(\\rho, u)$ is the coaction of $H^0$ on the unital $C^*$-algebra $A$, that is, $u=1\\otimes 1^0$. We investigate the relation between $\\Pic(A^s )$, the ordinary Picard group of $A^s$ and $\\Pic_H^{\\rho^s}(A^s )$ where $A^s$ is the stable $C^*$-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.07724","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1512.07724","created_at":"2026-05-18T01:23:48.043932+00:00"},{"alias_kind":"arxiv_version","alias_value":"1512.07724v1","created_at":"2026-05-18T01:23:48.043932+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.07724","created_at":"2026-05-18T01:23:48.043932+00:00"},{"alias_kind":"pith_short_12","alias_value":"WQF7J5KXTQ5R","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_16","alias_value":"WQF7J5KXTQ5RBGLL","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_8","alias_value":"WQF7J5KX","created_at":"2026-05-18T12:29:47.479230+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/WQF7J5KXTQ5RBGLLEISPV26T3Z","json":"https://pith.science/pith/WQF7J5KXTQ5RBGLLEISPV26T3Z.json","graph_json":"https://pith.science/api/pith-number/WQF7J5KXTQ5RBGLLEISPV26T3Z/graph.json","events_json":"https://pith.science/api/pith-number/WQF7J5KXTQ5RBGLLEISPV26T3Z/events.json","paper":"https://pith.science/paper/WQF7J5KX"},"agent_actions":{"view_html":"https://pith.science/pith/WQF7J5KXTQ5RBGLLEISPV26T3Z","download_json":"https://pith.science/pith/WQF7J5KXTQ5RBGLLEISPV26T3Z.json","view_paper":"https://pith.science/paper/WQF7J5KX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1512.07724&json=true","fetch_graph":"https://pith.science/api/pith-number/WQF7J5KXTQ5RBGLLEISPV26T3Z/graph.json","fetch_events":"https://pith.science/api/pith-number/WQF7J5KXTQ5RBGLLEISPV26T3Z/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WQF7J5KXTQ5RBGLLEISPV26T3Z/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WQF7J5KXTQ5RBGLLEISPV26T3Z/action/storage_attestation","attest_author":"https://pith.science/pith/WQF7J5KXTQ5RBGLLEISPV26T3Z/action/author_attestation","sign_citation":"https://pith.science/pith/WQF7J5KXTQ5RBGLLEISPV26T3Z/action/citation_signature","submit_replication":"https://pith.science/pith/WQF7J5KXTQ5RBGLLEISPV26T3Z/action/replication_record"}},"created_at":"2026-05-18T01:23:48.043932+00:00","updated_at":"2026-05-18T01:23:48.043932+00:00"}